The "Nicolas-Serre code", $(a,b) \leftrightarrow t^{ n}$, is a bijection between $N\times N$ and those $t^{ n}$, $n$ odd, in $Z/2[t]$. Suppose $A_{ n}$, $n$ odd, in $Z/2[t]$ are defined by: $A_{ 1}= A_{ 5}= 0$, $A_{ 3}= t$, $A_{ 7}= t^{ 5}$, and $A_{ n+8}= t^{ 8} A_{ n} + t^{ 2} A_{ n+2}$. A lemma, Proposition 4.3 of [6], used to study the Hecke algebra attached to the space of mod $2$ level $1$ modular forms, gives information about the codes $(a,b)$ attached to the monomials appearing in $A_{ n}$. The unpublished highly technical proof has been simplified by Gerbelli-Gauthier. Our Theorem 3.7 generalizes Proposition 4.3. The proof, in sections 1-3, is a further simplification of Gerbelli-Gauthier's argument. We build up to the theorem with variants involving the same recurrence, but having different sorts of initial conditions. Section 4 treats the recurrence $A_{ n+16}= t^{ 16} A_{ n} + t^{ 4} A_{ n+4} + t^{ 2} A_{ n+2}$. Theorem 4.1, the analog to Theorem 3.7 for this recurrence, is used in [2] and [3] to analyze level 3 Hecke algebras. Finally we introduce a variant code, $(a,b) \leftrightarrow w^{ n}$ which is a bijection between $N\times N$ and those $w^{ n}$, $n \equiv 1,3,7,9 \bmod{ 20}$, in $Z/2[w]$. We then study the recurrence $A_{ n+80}= w^{ 80} A_{ n}+ w^{ 20} A_{ n+20}$, $n \equiv 1,3,7,9 \bmod{ 20}$, with appropriate initial conditions. Lemma 5.5, derived from the results of sections 1-3, is the precise analog of Proposition 4.3 for this code, this recurrence, and these initial conditions. It is used in [4] and [5] to analyze level 5 Hecke algebras.
Categories: math.NT